Let $(V, K)$ and $φ, ψ : V \rightarrow V$ be linear transformations.
Furthermore, we define the bracket of $φ$ and $ψ$ as
$[φ, ψ] := φ ◦ ψ − ψ ◦ φ$.
We say that two linear transformations commute if
$[φ, ψ] = 0$,
where 0 is the zero map. Show that if $[φ, ψ]$ commutes with $φ$, then $[φ^k , [φ, ψ]] = 0 ∀k ∈ N$ and $[φ ^k , ψ] = 0$. Here, $φ k$ denotes the composition of $φ$ k-times with itself.